Optimizing experimental design in neutron reflectometry

An approach for optimal experimental design of neutron reflectivity experiments using metrics derived from the Fisher information is presented. This is demonstrated on a range of systems including lipid bilayers and magnetic heterostructures, and it is shown that small (or large) changes to the experimental setup can result in drastically reduced experimental count times.


A.2 Maximin Optimisation
The presence of information content in individual parameters may hide low information content in a combination of parameters. Our approach chooses the (linear) combination of values that (locally) has the least information content and maximises it. Figure 1 (right) shows an ellipse representing the FI, and how the ellipse corresponds to the amount of information in each parameter. The eigenvector of the FI matrix with minimum eigenvalue corresponds to the minor (short) axis of the ellipse and will be the linear combination of values with the least information.
The details of the mathematics describing the information quantities in figure 1 are as follows: consider two pairs of nearby parameters, one represented by a point in 2D parameter space p and Figure 1: Shown is an error ellipse with a large relative error in a combination of parameters (the x + y combination). Although there is information in both x and y, there is relatively little in the x + y direction. Details: If we linearly transform the space on the left so that the error ellipse becomes the unit circle, the unit circle will be transformed into an inverted form of the error ellipse with sizes corresponding to the information content.
another by p + ∆p, where the magnitude of ∆p is small and equal to r. The information divergence between these points is given by If ∆p is in the direction of the x-axis we can get an approximation which we will call I x D (p p + ∆p) ≈ 1 2 r 2 g xx = I x Similarly, in the y direction we have I y = r 2 g yy /2, and in the 45 • directions, x + y and x − y, we have I x+y = 1 4 r 2 (g xx + g yy ) + 1 2 r 2 g xy = 1 2 I x + I y + r 2 g xy I x−y = 1 2 I x + I y − r 2 g xy From this we can see that the total information is the same, whether we use a basis of x and y, or of x + y and x − y: I x + I y = I x+y + I x−y But we also see that, if the off-diagonal entry of g (i.e., g xy ) is either very positive or very negative, the information about x + y or x − y might be very small.
The model was defined with three roughness parameters: the silicon/silicon oxide and silicon oxide/bilayer interfacial roughnesses and a bilayer roughness that was shared between the other interfaces (inner headgroup/inner tailgroup, inner tailgroup/outer tailgroup, outer tailgroup/LPS core and LPS core/solution). The silicon substrate layer was defined using the known SLD of silicon (2.07 × 10 −6Å −2 ). The silicon oxide and inner headgroup layers were defined using their known SLDs (3.41 × 10 −6Å −2 and 1.98 × 10 −6Å −2 respectively), with the thickness and hydration of each layer set as parameters. The inner and outer tailgroup layers were defined using separate thickness parameters but a shared hydration parameter. The SLDs for the two layers, ρ inner TG and ρ outer TG , were defined using an asymmetry parameter, α, and the known SLDs of the DPPC and LPS tailgroups (7.45 × 10 −6Å −2 and −0.37 × 10 −6Å −2 respectively).
The LPS core layer thickness and hydration were set as parameters, but the SLD was defined using the mole fraction of D 2 O from the bulk water SLD, denoted here as x, and the known SLDs of the LPS core in D 2 O, ρ core D2O , and H 2 O, ρ core H2O (4.20 × 10 −6Å −2 and 2.01 × 10 −6Å −2 respectively).

A.3.2 Kinetics
The 1,2-dipalmitoyl-sn-glycero-3-phospho-(1-rac-glycerol) (DPPG) monolayer model was defined using a slab representation: air, monolayer tailgroup (either hydrogenated or deuterated), monolayer headgroup, and finally the bulk water solution of given SLD, ρ water ; table 1 summarises the parameters of the model. All model interfacial roughnesses (air/tailgroup, tailgroup/headgroup and headgroup/water) were parameterised. The tailgroup (both hydrogenated and deuterated) and headgroup thicknesses were defined using a shared lipid area per molecule (APM) parameter and the equation d = V/A, where d is the layer thickness, V is the layer volume and A is the lipid APM.
The tailgroup and headgroup volumes were calculated from the volumes of their constituent components, as summarised in table 2. For the tailgroup volume, V TG , this was relatively straightforward but for the headgroups, we needed to account for the hydrating water molecules. We did this by first multiplying the known water volume by the headgroup bound waters parameter to obtain the extra water volume in the headgroups, V bound , and then added this to the individual fragment volumes We calculated the hydrogenated and deuterated monolayer tailgroup SLDs using the previously calculated tailgroup volume and equation ρ = Σb/V, where ρ is the layer SLD, Σb is the neutron scattering length (SL) sum for the layer and V is the layer volume. The SL sums of the hydrogenated and deuterated tailgroups, Σb hTG and Σb dTG respectively, were calculated from the total SL of each constituent fragment. The SLs of the individual elements of the fragments are given in table 3.
Like the tailgroups, the headgroup SLD was determined using the previously calculated headgroup volume, the SL sums of the constituent fragments, and equation ρ = Σb/V. However, as with the headgroup volume calculation, we needed to account for the hydrating water molecules. We did this by first calculating the mole fraction of D 2 O from the bulk water SLD, x, to get the average SL sum per water molecule.   Table 3: Neutron scattering lengths for the components of the tailgroup and headgroup fragments.

A.3.3 Magnetism
The experimental scale factor, level of instrument background and instrument resolution function used to fit the data were 1.025, 4 × 10 −7 and constant 2.8% dQ/Q respectively. The model was defined using a slab representation consisting of air, platinum, yttrium oxide, yttrium iron garnet (YIG) and yttrium aluminium garnet (YAG);